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Full text of "Mathematical And Physical Papers - Iii"

14       ON   THE EFFECT OP THE INTERNAL FKICTION OF FLUIDS
the rate of sliding. In the theory it is supposed that in general the pressure about a given point is compounded of a normal pressure, corresponding to the density, which being normal is necessarily equal in all directions, and of an oblique pressure or tension, altering from one direction to another, -which is expressed by means of linear functions of the nine differential coefficients of the first order of u, v, w with respect to %, yy z, which define the state of relative motion at any point of the fluid. TSTow in the special case considered above, if we confine our attention to one direction, that of the plane of xy> the total pressure referred to a unit of surface is compounded of a normal pressure corresponding to the. density, and a tangential pressure expressed by /-& dvjdz, which tends to reduce the relative motion.
In trie solution of equations (2), p always appears divided by p. Let /-& = ///>. The constant p may conveniently be called the indeoc of friction of the fluid, whether liquid or gas, to which it relates. As regards its dimensions, it expresses a moving force divided by the product of a surface, a density, and the differential coefficient of a velocity with respect to a line. Hence fjf is the square of a line divided by a time, whence it will be easy to adapt the numerical value of /*' to a new unit of length or of time.
3. Besides the general equations (2) and (3), it will be requisite to consider the equations of condition at the boundaries of the fluid. For the purposes of the present paper there will be no occasion to consider the case of a free surface, but only that of the common surface of the fluid and a solid. Now, if the fluid immediately in contact with a solid could flow past it with a finite, velocity, it would follow that the solid was infinitely smoother with respect to its action on the fluid than the fluid with respect to its action on itself. For, conceive the elementary layer of fluid comprised between the surface of the solid arid a parallel surface at a distance h, and then regard only so much of this layer as corresponds to an elementary portion dS of the surface of the solid. The impressed forces acting on the fluid element must be in equilibrium with the effective forces reversed. Now conceive Ti to vanish compared with the linear dimensions of dS, and lastly let dS vanish*. It is evident that the conditions
.     * To be q.uite precise it would be necessary to say, Conceive U and dS to vanish together, 7i vanishing compared with the linear dimensions of dS; for so long as dS